\(A=111...1=\frac{10^{2n}-1}{9}\) ; \(B=444...4=4\times111...1=4\left(\frac{10^n-1}{9}\right)\)
\(\Rightarrow A+B+1=\frac{10^{2n}-1}{9}+\frac{4.10^n-4}{9}+1=\frac{10^{2n}+4.10^n+4}{9}=\left(\frac{10^n+2}{3}\right)^2\)
Do \(\left\{{}\begin{matrix}10^n\equiv1\left(mod3\right)\\2\equiv-1\left(mod3\right)\end{matrix}\right.\) \(\Rightarrow10^n+2⋮3\Rightarrow\frac{10^n+2}{3}\in N\)
\(\Rightarrow\left(\frac{10^n+2}{3}\right)^2\) là SCP hay \(A+B+1\) là SCP